
\(d=1\) . Let \(g(e)=e(k)\) . Then \(\mathbb {E}_e[Y_1]=\mathbb {E}_e[\tau _1(e)]\mu (k)\) . If \(k\in [\![-K,0]\!]\) , then the biased random walk \(Z_{\tau _n(e)}=\sum _{m=1}^nY_m\)  is transient (visits a neighbourhood of 0 finitely many times), from which we deduce that \(S_n^{{{\scriptscriptstyle {({k}})}}}=Z_n\)  itself is transient. If \(k\in [\![1,K]\!]\) , then \(\mathbb {E}[Y_1]=0\)  and, by the Chung-Fucks Theorem (see for example [1]}), the random walk \(Z_{\tau _n(e)}\)  is recurrent. It follows from Proposition REF  that \(S_n^{{{\scriptscriptstyle {({k}})}}}\)  visits 0 infinitely often.

\(d=2\) . Let \(g(e)=(e(k),e(\ell ))\) , \(k,\ell \in [\![1,K]\!]\) . Here again \(\mathbb {E}[Y_1]=0\)  and, by the Chung-Fucks Theorem, the 2-dimensional random walk \(Z_{\tau _n(e)}\)  is recurrent, and, by Proposition REF , so is \((S_n^{{{\scriptscriptstyle {({k}})}}},S_n^{{{\scriptscriptstyle {({\ell }})}}})\) .

\(d\ge 3\) . Let \(g\)  be the projection on a given \(d\) -dimensional space. Then, the transience of \(Z_n\)  follows from the transience of the \(d\) -dimensional random walk \(Z_{\tau _n(e)}=\sum _{m=1}^nY_m\) .

